#487674

order	512
nilpotency class	4
isoclinism	isoclinic class of #487635
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#534	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,43)(31,44)(33,45)(34,40)(35,41)(36,42)(37,46)(38,47)(39,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,23,33)(18,24,28,39)(20,31,21,30)(22,32,36,48)(25,38,26,37)(27,29,42,45)(34,47,35,46)(40,44,41,43);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,23,21)(18,25,28,26)(19,30,33,31)(22,35,36,34)(24,37,39,38)(27,40,42,41)(29,43,45,44)(32,47,48,46);(2,8)(3,10)(6,9)(11,16)(17,21,23,20)(18,26,28,25)(19,31,33,30)(22,35,36,34)(24,38,39,37)(27,41,42,40)(29,44,45,43)(32,47,48,46)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394870);